While that's a huge amount of money, buying a ticket is still
probably a losing proposition.

Consider the expected value

When trying to evaluate the outcome of a risky, probabilistic
event like the lottery, one of the first things to look at is
expected
value. The expected value of a randomly decided process is
found by taking all of the possible outcomes of the process,
multiplying each outcome by its probability, and adding all of
these numbers up. This gives us a long-run average value for our
random process.

Expected value is helpful for assessing gambling outcomes. If my
expected value for playing the game, based on the cost of playing
and the probabilities of winning different prizes, is positive,
then
in the long run the game will make me money. If expected
value is negative, then this game is a net loser for me.

Powerball and similar lotteries are a wonderful example of this
kind of random process. As of October in Powerball,
five white balls are drawn from a drum with 69 balls, and one red
ball is drawn from a drum with 26 balls. (As an aside, that rule
change is why prizes can get as big as the latest record: The
probability
of winning the jackpot is much lower than it used to be.)

Prizes
are then given out based on how many of a player's chosen
numbers match the numbers written on the balls. Match all five
white balls and the red Powerball, and you win the jackpot. In
addition, several smaller prizes are won for matching some subset
of the drawn numbers.

Powerball's website helpfully provides a list of the
odds and prizes for each of the possible outcomes. We can use
those probabilities and prize sizes to evaluate the expected
value of a $2 Powerball ticket. Take each prize, subtract the
price of our ticket, multiply the net return by the probability
of winning, and add all those values up to get our expected
value:

Business Insider/Andy Kiersz

At first glance, we seem to have a positive expected value at
$1.06. The situation, however, is more complicated.

Annuity versus lump sum

Our first problem is that the headline $800 million grand prize
is paid out as an annuity: Rather than getting the whole amount
all at once, you get the $800 million spread out in smaller — but
still multimillion-dollar — annual payments over 30 years. If you
choose to take the entire cash prize at one time instead, you get
much less money up front: The cash-payout value at the time of
writing is $496 million.

Looking at the lump sum, our expected value drops dramatically to
just $0.02:

Business Insider/Andy Kiersz

The question of whether to take the annuity or the cash is
somewhat nuanced. Powerball points out on
its FAQ site that in the case of the annuity, the state
lottery commission invests the cash sum tax-free, and you pay
taxes only as you receive your annual payments, whereas with the
cash payment, you have to pay the entirety of taxes all at once.

On the other hand, the state is investing the cash somewhat
conservatively, in a mix of various US government and agency
securities. It's quite possible, though risky, to get a larger
return on the cash sum if it's invested wisely. Further, having
more money today is frequently better than taking in money over a
long period of time, since a larger investment today will
accumulate compound interest more quickly than smaller
investments made over time. This is referred to as the time
value of money.

Taxes make things much worse

As mentioned above, there's the important caveat of taxes. While
state income taxes vary, it's possible that combined state,
federal, and in some jurisdictions local taxes could take as much
as half of the money. Factoring this in, if we're taking home
only half of our potential prizes, with the headline annuity
payout we have an expected value of -$0.47, making our Powerball
"investment" a bad idea:

Business Insider/Andy Kiersz

The hit to halving the cash one-time prize is similarly
devastating:

Business Insider/Andy Kiersz

When you consider issues like taxes and their effect on the
expected value of a Powerball ticket, you can see that the
lottery is a pretty bad "investment."

So when does it make sense to buy a ticket?

Well, probably never. But for fun, let's consider when we'd
actually get a breakeven expected value of zero or higher on our
"investment."

We need the expected value contribution of the jackpot to counter
the very negative and very likely outcome that we win nothing,
offset by the contributions from the smaller prizes. Adding up
all the non-jackpot prizes and our highly likely outcome of just
losing our $2, we need the jackpot expected value contribution to
be about $1.68:

Business Insider/Andy Kiersz

So, we need to solve the equation: Jackpot x (1 in 292,201,338) =
$1.68. Divide both sides by that insanely low probability that we
win the jackpot and we get our desired take-home winnings to be
$490,933,821.72.

Under our assumption that we could lose up to half of our
winnings to taxes, that means we need the pretax prize to be
$981,867,643.44.

Since we are inclined to take the lump sum rather than the
annuity, factoring in the importance of time value of money, if
we assume the same ratio between the annuity and the lump sum
that currently holds — a $800 million headline prize compared
with a $496 million cash lump-sum prize — that means that, to
even get close to a breakeven value, we would need the headline
annuity prize to be a whopping
$1,583,657,489.42.

Of course, even at that astonishingly high, nearly $1.6 billion
headline prize, we could run into the problem of the possibility
of splitting that prize with another winner. Factoring in further
issues with the lottery then, we would need a prize at least
twice as high as the current record to even consider buying a
ticket, and even then it's still probably a bad idea.